The end of June marks the end of Boris Johnson's Story of London festival, as Londoners may, or regrettably may not, know. Described by City Hall as a "truly pan-London" and "glorious" celebration of the capital's "past, present and future" including "hundreds of events", it was only patchily promoted and sometimes very tricky to locate.

My search for SoL began at the start of the month when its website listed an exhibition in Whitechapel that turned out to be closed and another, in Canary Wharf, that was either non-existent or so difficult to find it might as well have been. It ended on Saturday when my plan to partake of SoL's Lives of Buildings weekend by visiting an exhibition foundered on an encounter with a security guard in Clerkenwell. "It's only open on weekdays," he explained.

I've not been alone in such woes. A woman from Hounslow called Helen who reads my Guardian blog about London has been in frequent contact with stories of boroughs that couldn't contribute to Mayor Boris's history jamboree because they were told about it too late, and of Tourist Information offices, including Heathrow's, that hadn't been told about it at all.

All this is such a shame, because when I've found a SoL component, it's been good. At King's Place, I saw a predictably excellent talk on London's rail travel history by the writer Christian Wolmar, followed by an instructively Tory account of the capital's blitz experience by the historian Andrew Roberts. Stepping out of my cultural comfort zone, I watched a choir perform Orlando Gibbons's Cries of London on the street at Spitalfields. The Big Smoke, a BFI compilation of documentary clips from the late 19th century to VE Day, found its way to my neighbourhood St John Ambulance hall, in Hackney.

Another correspondent went with his family to one of the festival's setpiece specials, a Tudor joust at Eltham Palace. "Not bad, if you like jousting," he said. But his account also compounded the inescapable sense that the SoL has been cobbled together on the cheap – and suffered as a result.

Should Boris hold his hand up? He promoted the festival with two high-profile press conferences, one at Hampton Court (with a man dressed as Henry VIII) and another at the Tower (with Beefeaters) – but his budget didn't stretch to many posters around town, a special brochure in Time Out, or, it would seem, sufficient human resources to ensure correct website information.

The mayor has talked up the recession-beating properties of the capital's "cultural offer", but his paring of GLA spending suggests underinvestment in the SoL's contribution. Attendance at those King's Place talks was in the low 20s: not many, even on a Sunday morning. At Spitalfields, punters were outnumbered by choristers. "There wasn't any publicity," one said.

Many who voted for Johnson would think frugality apt, and Munira Mirza, his director of culture, may find partner institutions a more fruitful source of additional funding next year. An approving view of the SoL might see it as exemplifying both Johnsonian parsimony and a determination to restore a traditionalist and universalist approach to British history that, in his view, has been sacrificed to multiculturalism for too long. Mirza denies the claims of harsher critics that the SoL has been staid, elitist, in some instances too expensive or largely an ineffective exercise in re-marketing attractions that existed anyway. For me, though, the full potential of a good idea has simply yet to be fulfilled.

Big cities can thrive on big debates about themselves, and future SoLs should strive to promote one. New Yorkers have a powerful sense of their home town's past and character, one that embraces newcomers and those to the Big Apple born. Groovy Barcelona self-describes with art and monuments. Paris fusses over its appearance constantly. Rome just stands there being Roman. London tells its own story drawing in its way on all these techniques, but its internationalism – nearly half its inhabitants of working age were born abroad – its government's complex federalism and the sheer vastness of the place make it especially difficult to capture in coherent narrative.

It would go against the grain with Boris to increase the mayoral subsidy or take a more top-down or didactic approach to the SoL, but perhaps he should re-think. His love of history is deep and his populist gifts considerable. Leading a big conversation about the capital's sense of itself is fully consistent with the job of mayor, which is often more about talking loud and persuading than exercising the post's limited powers. With the Olympics approaching and the world looking our way, there is no better time for Boris to think bigger, be bossier and make more boldly his case for how the Story of London should be understood and told. I'd probably disagree with most of it, but that's OK. What is history if not a political background?